RESOLUTION-CONTROLLED CONDUCTIVITY ...winkler/media/res_eit.pdfCEM in [14]. For σ,eσ∈L ∞ + (Ω) and considering Rσ as an operator on L2(∂Ω), we call λσ,eσ = R σ,L −Reσ,L

RESOLUTION-CONTROLLED CONDUCTIVITY DISCRETIZATION

IN ELECTRICAL IMPEDANCE TOMOGRAPHY

ROBERT WINKLER AND ANDREAS RIEDER

Abstract. This work contributes to the numerical solution of the inverse problem of deter-mining an isotropic conductivity from boundary measurements, known as Electrical ImpedanceTomography. To this end, we first investigate the imaging resolution of the Complete ElectrodeModel in a circular geometry using analytic solutions of the forward problem and conformalmaps. Based on this information we propose a novel discretization of the conductivity spacewhich explicitly depends on the electrode sizes and locations. Roughly speaking, the result-ing conductivity meshes comply with the maximal resolution provided by discrete data with aknown noise level. We heuristically extend this approach to domains of arbitrary shape andpresent its performance under a Newton-type inversion algorithm.

1. Background and motivation

Electrical impedance tomography (EIT) is an imaging method for determining the electricalconductivity of an object from measurements on its surface. The underlying mathematicalmodel is an elliptic boundary value problem where the Dirichlet data represents the potentialon the object surface, the Neumann data represents the normal current flow through the surfaceand an elliptic PDE models the flow of current inside the object. The coefficient of the PDE isthe searched-for conductivity. The evaluation of the Neumann-to-Dirichlet (ND) operator fora given conductivity is called the forward problem of EIT. The task in EIT is usually to solvethe inverse conductivity problem (ICP) of determining the conductivity from the knowledge ofthe ND operator.

Electrode models and inversion. The pioneering work of Calderon [6] started the investi-gation of the continuum boundary model, where a complete knowledge of the continuum NDoperator is assumed. However in practice, one can only apply currents and measure potentialsthrough finitely many electrodes. As a consequence, the available boundary data is finite-dimensional – the ND map of this discrete model is a matrix. Moreover, due to electrochemicaleffects and the conducting nature of electrodes, it is impossible to prescribe Neumann dataor access Dirichlet data explicitly anywhere at the boundary. Instead, the accurate Complete

Electrode Model (CEM) [29] described in section 2.2 uses Robin-type boundary conditions toaccount for these effects.

Analytic properties of electrode models are rare. [19] show the injectivity of the Frechetderivative of the ND map for piecewise polynomial conductivities on a triangulation of thedomain for the CEM. The number of electrodes necessary for injectivity is finite, but unknownfor any fixed triangulation. [4] use a model reduction approach to transform the ICP into theproblem of determining the resistors in a network, which is uniquely solvable under certainconditions. However, the reduction approach is theoretically justified for radially symmetricnear-constant conductivities only and the boundary model is slightly different from commonlyused electrode models.

Stability of the reconstruction scheme is a critical matter for solving the ICP with measureddata. Due to Alessandrini’s logarithmic stability estimate [1, Theorem 1] in the operator norm

2010 Mathematics Subject Classification. 65N20, 65F22, 92C55.The work of the authors was supported by the German Research Foundation (DFG) under grant RI 975/7-1.

1

2 ROBERT WINKLER AND ANDREAS RIEDER

of the Dirichlet-to-Neumann (DN) map, the ICP for the continuum model is usually said to beexponentially unstable. Later results (e.g. [12], [24], [20]) show that the stability decays rapidlyaway from the boundary in typical norms. The consequence of all results is that regularizationis necessary to achieve a stable inversion scheme for the ICP. In the absence of a general analyticresult, stability studies for electrode models are usually done by investigating what can actuallybe detected in a fixed domain geometry and electrode setting. Here the notion of sensitivity (ordistinguishability [15], detectability), i.e. the effect of perturbations in conductivity to boundarymeasurements, is important.

The decay of sensitivity away from the boundary leads to a decrease of conductivity imagingresolution. This has been considered for numerical inversion by coarsening the conductivitydiscretization away from the boundary, often with additional constraints. For example, [20] usea distinguishability criterion for the continuum model (see section 5.3), while [7, 3] match thenumber of conductivity coefficients to the degrees of freedom available in the ND map (the lattermotivated by an analogy of EIT to resistor networks, see section 5.4). An explicit resolution-

based quantification of the discretization size for the CEM across the domain is an open taskthat we will tackle in this work.

Most algorithms for the computation of the ICP are either motivated by the continuum model(for example complex geometrical optics solutions [22], factorization method [5]) or regularizediterative methods for an electrode model. An overview can be found in [21]. In this work weconsider the iterative Newton-type method REGINN for the CEM [18].

Aim and structure of this work. We have pointed out that it is necessary to discretize theconductivity space and to find a suitable regularization scheme when solving the ICP numericallyfor an electrode model and noisy measurements. While the former is traditionally done bya simple triangulation of the domain, the choice of regularization is delicate as it imposesassumptions on the searched-for conductivity – usually on its smoothness – to control noiseamplification during inversion. As a consequence, EIT images are typically heavily blurred.

Our tailored discretization scheme for the conductivity space relies on a novel analytic forwardsolution of the CEM in the presence of a perturbation in conductivity. It is piecewise constanton a partition of the domain with a locally adaptive, noise-dependent mesh size which we call anoptimal resolution mesh. It explicitly inherits resolution information by modelling the smallestdetails which are recoverable from boundary data with a given noise level. Solving the ICP onthese meshes with a Newton-type algorithm can be done efficiently with the estimated noiselevel as the only free design parameter for regularization.

In section 2, we introduce the relevant EIT models and notations. In section 3, we derivean analytic solution for the CEM forward problem on a disk in the presence of a centeredcircular perturbation in conductivity. Next, we use conformal maps to extend this result toperturbations at arbitrary locations in section 4. We use these analytic solutions to get localresolution information of the CEM. With this information, we design optimal resolution meshesto discretize the conductivity space in section 5. We outline the connections between the CEMand the continuum model and also compare our results to the results of [20] and the meshesarising from resistor networks in [4]. We then heuristically extend our results to domains ofarbitrary shape. Finally, we demonstrate the performance of these discretizations using theregularized inexact Newton scheme REGINN [25] in section 6.

2. Preliminaries

A potential u ∈ H1(Ω) on a source-free, simply connected domain Ω ⊂ R2 with piecewise

Lipschitz boundary is a solution of the elliptic equation

−∇ · (σ∇u) = 0 on Ω,(1)

where σ ∈ L∞+ (Ω) = ϕ∈L∞(Ω)|ϕ≥ c> 0 a.e. for some c∈R>0 is the isotropic conductivity

coefficient.

RESOLUTION-CONTROLLED DISCRETIZATION IN EIT 3

2.1. Continuum boundary model. For given Dirichlet boundary data

u|∂Ω = f ∈ H1/2(∂Ω),(2)

the problem (1),(2) has a unique solution. Denoting by ν the outer normal on ∂Ω, we call

σ∂u

∂ν=: iν ∈ H

−1/2⋄ (∂Ω) =

ϕ ∈ H−1/2(∂Ω) |〈ϕ, 1〉∂Ω = 0

(3)

the corresponding Neumann data. Conversely, the problem (1),(3) has a solution which isunique if we require the trace of u to have vanishing mean, that is

u|∂Ω ∈ H1/2⋄ (∂Ω) =

ϕ ∈ H1/2(∂Ω) |〈ϕ, 1〉∂Ω = 0

.

Hence, the ND operator

Rσ : H−1/2⋄ (∂Ω) → H

1/2⋄ (∂Ω), iν 7→ f,

and its inverse are well-defined and one-to-one. The ICP in this setting is the inversion of theoperator σ 7→Rσ. Uniqueness is shown in [2].

2.2. Electrode boundary models. Electrode boundary models describe the injection of cur-rents and the measurement of potentials through a finite number of electrodes. Assume we haveelectrodes1

E1, . . . , EL ⊂ ∂Ω, L ∈ N≥2,

at which we can inject current patterns I ∈RL⋄ =

V ∈R

L |〈V,1〉=0and measure the resulting

potential vectors U∈RL⋄ . We assume that normal current iν at the domain boundary only occurs

at electrodes and that the electrodes are perfect conductors, i.e. the potential is constant oneach electrode. Furthermore, we model contact impedances zl≥ 0 that occur at each electrode-domain interface and cause a potential drop zliν , cf. [29]. This leads to Robin-type boundaryconditions

iν = 0 on ∂Ω \E, E = E1 ∪ . . . ∪ EL,(4a)

u+ zliν = Ul on El, l = 1, . . . , L,(4b)∫

El

iν dS = Il, l = 1, . . . , L.(4c)

For zl ≡ 0, (1),(4) is called the Shunt Model, for zl > 0 it is called Complete Electrode Model

(CEM). The CEM is usually defined weakly as the unique solution (u,U)∈H1⋄ (Ω)×R

L⋄ of

a( (u,U), (w,W ) ) =

L∑

l=1

IlWl for all (w,W ) ∈ H1(Ω)× RL⋄ ,(5)

where a :(H1(Ω)×R

L⋄

)×(H1(Ω)×R

L⋄

)→R,

a ( (v,W ), (w,W ) ) =

∫

Ωσ∇v · ∇w dx+

L∑

l=1

1

zl

∫

El

(v − Vl)(w −Wl) dS,(6)

see [29]. We denote the ND (or Current-to-Voltage) map of the CEM by

Rσ,L : RL⋄ → R

L⋄ , I 7→ U.(7)

The ICP for the discrete setting is the inversion of the operator σ 7→Rσ,L with σ restricted tosome subspace of L∞

+ (Ω). In practice, Rσ,L is given in a certain measurement basis of M ∈N

measurements, that is, we know current vectors I(m) ∈RL⋄ and their corresponding potentials

U (m) ∈RL⋄ satisfying Rσ,LI

(m)=U (m) for m=1,...,M . Since all potentials have vanishing mean,we have rank(Rσ,L)≤L−1. Usually, M =L−1 or M =L.

1We identify electrodes with the surface they cover on the domain boundary.


2.3. Sensitivity for detecting perturbations. A keystone in our analysis is the investigationof the sensitivity of measurements to perturbations in conductivity.

This is a common method for analyzing EIT models and has been done e.g. in [15] for thecontinuum boundary model and, very recently, for deriving resolution verification tests for theCEM in [14].

For σ,σ ∈L∞+ (Ω) and considering Rσ as an operator on L2(∂Ω), we call

λσ,σ =

∥∥Rσ,L −Rσ,L

∥∥2∥∥Rσ,L

∥∥2

and λ∗σ,σ =

‖Rσ −Rσ‖2‖Rσ‖2

(8)

the (relative) sensitivities for distinguishing σ from σ by electrode and continuum boundarymeasurements, respectively. Of special interest is the sensitivity

λσ := λσ,1

for distinguishing a perturbed conductivity σ(x)=1+η(x) from the homogeneous case. Assumethat we have a measurement setting with the relative spectral error bound

‖Rmeas −R1,L‖2‖R1,L‖2

≤ ε, ε > 0,(9)

where Rmeas is a noisy measured ND map of the CEM in the homogeneous case σ≡ 1. Then,it is natural to call a perturbation η detectable in this setting if λσ >ε. The size |supp(η)| of aperturbation η reaching the resolution limit of the measurement setup, i.e. resulting in λσ = ε,strongly depends on the contrast of the perturbation and its location inside the domain.

Remark: Our definition of sensitivity (8) and spectral error (9) is slightly different from [14]and [15, sec. III], where the absolute error ‖Rσ−Rσ‖2 is considered. This absolute error isstrongly dependent on the underlying background conductivity – and the contact impedancein a corresponding CEM version – and corresponds to an absolute spectral measurement noiselevel, independent of the magnitude of the measured potentials. In contrast, our definition isnormalized by the maximum singular value of the ND map, which corresponds to a measurementerror relative to the magnitude of the measured potentials when applying normalized currents.It is less sensitive to the underlying background conductivity and the contact impedances. Wetherefore believe that it is better suited for quantifying the sensitivity and resolution of a givenmeasurement geometry.

3. An analytic solution of the CEM forward problem for a centered circular

perturbation in conductivity

The forward problem in EIT is usually solved with numerical methods like finite elements.For some basic geometries, analytic solutions exist in terms of Fourier series. We present ananalytic method to compute the operator Rσ,L of the CEM for a homogeneous disk containing asingle centered circular perturbation. That way, we can compute the sensitivity λσ for detectingthese circular perturbations explicitly. This is achieved by merging and generalizing the ansatzof [29, Appendix 3] for a single centered perturbation with the model of [11]. By extending theCEM to contact impedances that are varying along each electrode in Appendix B, we set stagefor exploiting conformal maps and considering perturbations at arbitrary locations in section 4.

Let Ω⊂R2 be the unit disk2 and let, for some r0∈ (0,1), the conductivity in polar coordinates

be given as

σ(r, θ) =

σ0, 0 ≤ r ≤ r0,

σ1, r0 < r ≤ 1.

2All calculations can easily be extended to disks of arbitrary radius.


For iν ∈L2(∂Ω), the Fourier representations of a Dirichlet-Neumann pair on ∂Ω are derived inAppendix A.1 as

f(θ) = u(1, θ) = u0 +

∞∑

k=1

ak cos(kθ) + bk sin(kθ),(10a)

iν(θ) = σ1∂u

∂r(1, θ) = σ1

∞∑

k=1

dk (ak cos(kθ) + bk sin(kθ)) ,(10b)

for some Fourier coefficients u0,ak,bk ∈R and

dk = k1− ck1 + ck

, ck =σ1/σ0 − 1

σ1/σ0 + 1r2k0 , k ∈ N.(11)

This means that for centered perturbations, the trigonometric functions are eigenfunctions ofRσ with eigenvalues τσ,k =(σ1dk)

−1.To determine these coefficients for a given electrode potential vector U in the CEM, we

generalize the approach of [11, Appendix]. For the CEM and σ smooth near ∂Ω, we have

u|∂Ω ∈ H3/2−α⋄ (∂Ω) and iν ∈ H

1/2−α⋄ (∂Ω) for all α > 0, see [9, Remark 1] and the references

therein. This guarantees a sufficiently fast decay of the coefficients and a convergence of theabove series. Once the Fourier coefficients are known, we compute the resulting current vectorI by (4c). These calculations are carried out in Appendix A.2. Note also the remark thereinregarding the truncation of the Fourier series for practical computation. Repeating this processfor L−1 basis vectors U (1),...,U (L−1) of RL

⋄ , we obtain the DN map R−1σ,L and we can determine

the ND map Rσ,L for the perturbed conductivity σ and finally the sensitivity λσ by (8).Clearly, λσ depends (nonlinearly) on the conductivities σ0 and σ1 and on the radius r0 of the

perturbation. Moreover, the sensitivity is monotonous in the following sense: Let η > 0 and let

σB = σ1 + ηχB , σD = σ1 + ηχD, B ⊂ D ⊂ Ω.(12)

Then, λσD ,σ1≥λσB ,σ1

which is shown in Appendix C. It is a CEM version of [13, Appendix I]. Bythe same argument, a similar monotony holds for increasing values of η and fixed perturbationsize.

When studying the imaging resolution of EIT settings, it is insightful to know the smallestradius r0 of a circular perturbation centered about the origin O inside the unit disk that can bedetected from measurements with spectral noise level ε. This can be achieved by determiningr0 such that λσ = ε for σ=1+ηχBr0

(O). Due to the above monotonicity, the smallest radius r0for conducting perturbations is reached for η→∞.3

4. The CEM under conformal mapping

Conformal maps are angle-preserving deformations of the plane. In particular, solutions of theelliptic PDE (1) remain valid in a conformally transformed geometry (see standard literature,e.g. [23, 26]).

Conformal maps have been used for analyzing the continuum boundary model of EIT on theunit disk e.g. in [28] and for investigating its resolution in [27]. Moreover, quasi-conformal mapshave been used for reconstructing quasi-conformal (anisotropic) images of realistic boundaryshape settings e.g. in [16] for continuum boundary and electrode models.

Using conformal diffeomorphisms on the unit disk, we will compute the ND map of theComplete Electrode Model analytically for circular perturbations at arbitrary locations of theunit disk by reducing the non-centered situation to the centered case of the previous section.This can be done by mapping the disk conformally onto itself such that the non-centeredperturbation is centered about the origin, then transforming the CEM boundary descriptionaccordingly and solving the forward problem in this transformed setting.

3In our numerical calculations, we set η=106−1.


We will show that due to the nature of conformal maps, the resulting ND map agrees with theone of the initial geometry when the boundary settings, in particular the electrode parameters,are modified accordingly. With this technique, we can determine λσ for all circular perturbationsinside a disk.

4.1. Conformal mapping of the CEM on the unit disk. Let w be a conformal diffeo-morphism on the unit disk that maps a point R= (R,ϕ) ∈ [0,1)× [0,2π) to the origin O. For4

C∼R2, w and its inverse are given (up to rotation) by5

z 7→ w(z) =e−iϕz −R

1−Re−iϕzand y 7→ w−1(y) = eiϕ

y +R

1 +Ry.(13)

In particular, a point on the boundary is again mapped to the boundary since∣∣∣w(eiθ)

∣∣∣ =∣∣∣w−1(eiθ)

∣∣∣ = 1, θ ∈ [0, 2π).(14)

Our aim is to transform an EIT setting conformally such that a perturbation on a disk BrQ(Q),Q=(Q,ϕ)∈ [0,1)× [0,2π) and 0<rQ< 1−Q, is mapped by w to a disk Br0(O) centered aboutthe origin for some 0< r0 < 1. The relations between the perturbation parameters Q and rQ,the parameter R of the according conformal map w and the radius r0 are given by

Q = R1− r20

1− r20R2

and rQ = r01−R2

1− r20R2,(15)

that is

R =1 +Q2 − r2Q −

√(1 +Q2 − r2Q)

2 − 4Q2

2Qand

r0 =1−Q2 + r2Q −

√(1−Q2 + r2Q)

2 − 4r2Q

2rQ,

which can be seen by plugging ±r0 into w−1, using the axis of symmetry in direction ϕ andthe fact that w−1 maps circles onto circles. Note that r0 > rQ and R>Q. The action of w isdepicted in Figure 1.

w

O r0

ϕ

QRrQ

Figure 1. The conformal map w maps B1(O) onto itself, R to O and BrQ(Q)onto Br0(O).

4We switch between polar coordinate notation in R2 and complex notation, using whatever is more convenient.

5The arguments (R,ϕ) are omitted. They denote the geometric parameters of w and w−1 throughout thiswork.


To parametrize the transformed boundary, let

ϑ := g(θ) := arg(w(eiθ)), θ = g−1(ϑ) = arg(w−1(eiϑ)) and set

wr := |w| , w−1r :=

∣∣w−1∣∣ , wθ := arg(w), w−1

ϑ := arg(w−1).

Being conformal maps, w and w−1 satisfy the Cauchy-Riemann equations. In particular,

g′(θ) =∂wθ

∂θ

∣∣∣∣r=1

=∂wr

∂r

∣∣∣∣r=1

=1−R2

1 +R2 − 2R cos(θ − ϕ)and

(g−1)′(ϑ) =∂w−1

ϑ

∂ϑ

∣∣∣∣∣r=1

=∂w−1

r

∂r

∣∣∣∣r=1

=1−R2

1 +R2 + 2R cos(ϑ).

(16)

Moreover, it follows from (14) that

0 =∂wr

∂θ

∣∣∣∣r=1

=∂w−1

r

∂ϑ

∣∣∣∣r=1

= − ∂wθ

∂r

∣∣∣∣r=1

= − ∂w−1ϑ

∂r

∣∣∣∣∣r=1

.

When transforming a potential u defined on the unit disk conformally to uw by uw(w(z))=u(z),i.e. uw(z)=u(w−1(z)), the w-transformed Dirichlet data on the unit circle reads

fw(ϑ) = uw(1, ϑ) = u(w−1(1, ϑ)) = f(g−1(ϑ)) = f(θ).

For σ ≡ σ1 near the boundary and using the chain-rule in polar coordinates, the transformedNeumann data reads

iwν (ϑ) =σ1∂uw

∂r(1, ϑ)

= σ1∂u

∂r(w−1(r, ϑ))

∂w−1r

∂r(r, ϑ)

∣∣∣∣r=1

+ σ1∂u

∂θ(w−1(r, ϑ))

w−1ϑ

∂r(r, ϑ)

∣∣∣∣∣r=1︸︷︷︸

=0

=1−R2

1 +R2 + 2R cos(ϑ)iν(g

−1(ϑ)) =1−R2

1 +R2 + 2R cos(ϑ)iν(θ).

(17)

We will now transform the boundary conditions (4) such that the ND map does not changeunder conformal mapping which is crucial for our further analysis. In particular, the position,width and contact impedance of each electrode change. Condition (4a) states that normalcurrent only occurs on electrode surfaces. For

Ewl := g(El),

it is obvious that (4a) holds in the transformed setting. Recalling (17), condition (4c) is readilychecked as

Il =

∫

El

iν(θ) dθ =

∫

g(El)iν(g

−1(ϑ))(g−1)′(ϑ) dϑ

=

∫

Ewl

iwν (ϑ)

(∂w−1

r

∂r(r, ϑ)

∣∣∣∣r=1

)−1 (g−1)′(ϑ) dϑ = Iwl .

The last equality holds due to property (16). The transformation of condition (4b) requires atransformation of the contact impedances. We have that

Ul =f(θ) + zliν(θ) = f(g−1(ϑ)) + zliν(g−1(ϑ))

=fw(ϑ) + zl1 +R2 + 2R cos(ϑ)

1−R2iwν (ϑ)

!= fw(ϑ) + zwl (ϑ)i

wν (ϑ) = Uw

l ,


thus set

zwl (ϑ) :=1 +R2 + 2R cos(ϑ)

1−R2zl.(18)

The resulting contact impedances are functions of the angular variable ϑ, i.e. they are no longerconstant. The CEM in the w-transformed setting giving rise to the same ND map as (4) reads

iwν = 0 on ∂Ω \ Ew, Ew = Ew1 ∪ . . . ∪ Ew

L ,(19a)

fw(ϑ) + zwl (ϑ)iwν (ϑ) = Ul on Ew

l , l = 1, . . . , L,(19b)∫

Ewl

iwν dS = Il, l = 1, . . . , L.(19c)

For centered circular perturbations, the ND map can again be computed analytically. This iscarried out in Appendix B.

Remark: The scaling of the complex plane z 7→ αz, α> 0, is a conformal map. The normalcurrent density resulting from a transformed potential is iν 7→α−1iν . The corresponding bound-ary settings for preserving the ND map are El 7→αEl and zl 7→αzl. Thus, we can consider disksof arbitrary radii.

5. Optimal resolution meshes and approximations

In section 5.1, we propose a discretization of the conductivity space for the CEM in which wequantify the local resolution of the conductivity mesh explicitly using the analytic sensitivityinformation derived in the previous section. This works for any electrode configuration on theunit circle (cf. Figure 2). Due to the high computational cost, this method is mainly of theo-retical interest, and an approximation scheme is derived for quickly generating approximationsto these discretizations in section 5.5.

5.1. Optimal resolution meshes on the unit disk. With the conformal mapping technique,we are able to determine the sensitivity for detecting circular perturbations anywhere in a disk.With this information, we design a partition of the disk in which perturbations in each cell haveroughly the same impact on boundary measurements. To that end, we “fill” the disk with non-overlapping circular cells of fixed sensitivity ε > 0 and apply a Voronoi tessellation afterwardsto get a partition of the entire disk. We call the resulting partition an optimal resolution mesh.

The generation of such a mesh with approximate sensitivity ε for perturbations of conductivityσ0 in a background of conductivity σ1 thus goes in several steps (Algorithm 1):

First, choose a finite set of points T ⊂B1(O) which are candidates for centers of circular cells.Next, successively pick points Q from T and determine the radii rQ of perturbations centeredabout Q and resulting in sensitivity λσ = ε for σ = σ1 + (σ0 − σ1)χBrQ

(Q). The radii can be

found by a line-search strategy in very few steps since rQ depends locally quadratic on λσ. IfBrQ(Q) lies inside Ω and does not intersect with the previously covered area C ⊂Ω, add Q tothe set of valid points P and add BrQ(Q) to the covered area C. Otherwise, discard Q. Theresult is a union of non-overlapping circles C with centers P, each resulting in sensitivity λσ = εfor perturbations of conductivity σ0 in a homogeneous background medium of conductivityσ1. Finally, apply a Voronoi tessellation to P and restrict it to B1(O) to get a partition ofthe domain. Results for various uniformly and non-uniformly distributed electrode settings areshown in Figure 2.Remark: The set T should be chosen finer than the maximum expected resolution to achievegood results. To avoid big gaps between the circles, it is advisable to pick the innermost pointfrom the set T first and then successively pick points with the biggest distance to the boundary.A possible choice for T are points on concentric circles.


Algorithm 1: Generation of an optimal resolution mesh on Ω=B1(O)

Input: ε,σ0,σ1;

Choose a finite set of test points T ⊂B1(O);

Set C= ∅, P = ∅;repeat

Pick a point Q from T and set T := T \Q;

Find rQ> 0 such that λσ = ε for σ=σ1+(σ0−σ1)χBrQ(Q);

if BrQ(Q)⊂B1(0) and BrQ(Q)∩C= ∅ then

Set P :=P∪Q, C := C∪BrQ(Q),T :=T \BrQ(Q);

end

until T = ∅;Output: Voronoi tessellation of P, truncated to B1(O);

(a) ε=0.02, 182 cells (b) ε=0.01, 445 cells (c) ε=0.02, 177 cells

Figure 2. Circular perturbations (top) and according Voronoi tessellations(bottom) generated by Algorithm 1. Settings: σ1=1, σ0 =106 (conducting per-turbation) and contact impedances z≡ 0.01. (a), (b): 16 equally spaced elec-trodes that cover 50% of the boundary. (c): Nonuniform setting with 16 smallelectrodes on the upper half-circle and one big electrode at the bottom.

5.2. Comparison with the continuum boundary model. To compare the resolution achievedby the CEM with the continuum model, we now derive upper bounds for the sizes of perturba-tions resulting in sensitivity λ∗

σ ≥ ε in the continuum model.For a centered circular perturbation inside the unit disk, i.e. σ = σ1+(σ1−σ0)χBr0 (O), the

eigenvalues τσ,k of Rσ for the normalized eigenfunctions cos(k·)/√π, sin(k·)/√π, k ∈ N, are

derived in section 3. In the homogeneous case σ0 = σ1, we have ‖Rσ1‖2 = σ−1

1 . For conductingperturbations σ0 >σ1, we have −1<c1 <c2 < ... < 0 in the definition of τσ,k, thus

‖(Rσ −Rσ1)fk‖2 = |τσ,k − τσ1,k| ‖fk‖ =

∣∣∣∣1 + ck

σ1k(1− ck)− 1

σ1k

∣∣∣∣ =−2ck

σ1k(1 − ck)

is decreasing in k for eigenfunctions fk.The sensitivity for detecting a centered perturbation is given by

λ∗σ,σ1

=−2c1

(1− c1), c1 =

σ1/σ0 − 1

σ1/σ0 + 1r20.


By formally letting σ0 →∞ for a perfectly conducting inclusion, the radius r0 resulting in asensitivity λ∗

σ,σ1= ε is given as

r0 =

√ε

2− εfor ε < 1.

To investigate the resolution at a point Q∈B1(Ω), we use the conformal map w−1 that mapsBr0(O) onto BrQ(Q) for some rQ> 0. The parameters are given by (15) as

R = R(Q, r0) =r20 − 1 +

√(1− r20)

2 + 4Q2r202Qr20

and rQ = r01−R(Q, r0)

2

1− r20R(Q, r0)2.(20)

By (16), the Dirichlet data fk and the corresponding normal current iν,k are transformed to

fw−1

k (θ) = fk(g(θ)) and iw−1

ν,k (θ) =1−R(Q, r0)

2

1 +R(Q, r0)2 − 2R(Q, r0) cos(θ − ϑ)iν,k(g(θ)).

Denote by σw the w−1-transformed conductivity. Although we do not know the singular systemof Rσw explicitly, we can get a lower bound for λ∗

σw,σ1by setting k = 1, using the conformal

mapping properties Rσw iw−1

ν,1 = τσ,1fw−1

1 and Rσ1iw

−1

ν,1 = τσ1,1fw−1

1 and using the functions fw−1

1

and iw−1

ν,1 in (8). This leads to

λ∗σw,σ1

≥

∥∥∥(Rσw −Rσ1)iw

−1

ν,1

∥∥∥2

‖Rσ1‖2∥∥∥iw−1

ν,1

∥∥∥2

=|τσ,1 − τσ1,1|

σ1

∥∥∥fw−1

1

∥∥∥2∥∥∥iw−1

ν,1

∥∥∥2

= αwλ∗σ,σ1

, where αw =

∥∥∥fw−1

1

∥∥∥2∥∥∥iw−1

ν,1

∥∥∥2

.

Thus to find the radius rQ resulting in λ∗σw ,σ1

≥ ε=αwλ∗σ,σ1

in the perfectly conducting case, weneed to solve the implicit equation

r0 =

√α−1w ε

2− α−1w ε

.

It is implicit because the parameter R of the conformal map w−1 and thus αw depend on r0.However for α−1

w ε≪ 1, r20 depends almost linearly on ε and the solution can be found quicklyby the fixed-point iteration

r0,k+1 =

√α−1wkε

2− α−1wkε

, r0,0 =

√ε

2− ε,

where w−1k is the conformal map with parameter R=R(Q,r0,k).

Once r0 is found, we can explicitly compute AQ = πr2Q, an upper bound for the size of aperfectly conducting perturbation centered about Q and resulting in sensitivity λ∗

σw ,σ1≥ ε. In

particular for f1=cos(·)/√π, AQ→ 0 as Q→ 1 which is shown in Figure 3.Figures 3a–d show AQ as a function of Q for a fixed sensitivity in the continuum model

(λ∗σ =0.015) and in various CEM settings6 (λσ =0.015) with background conductivity σ1=1.

While the perturbation size AQ gets arbitrarily small near the boundary for the continuummodel, it is bounded away from 0 for the CEM because the concentration of Neumann datanear a point is limited by the distance and size of the electrodes. We observe that increasing thenumber of electrodes does not necessarily increase the resolution inside the disk (a). A decreaseof the contact impedances increases the resolution at the center (b). A decrease of the electrodesizes increases the resolution near an electrode, but decreases the resolution at the center (c).A comparison of different types of resistive and conducting perturbations, along with the theresults of [20] (see section 5.3), is shown in (d).

6The angle ϕ of the point Q=(Q,ϕ) is fixed such that Q approaches the center of an electrode as Q→ 1 inthe CEM.


It is worth mentioning that perfectly conducting ( in 3d) and isolating (H) perturbationsresult in roughly the same resolution. The resolution of low-contrast conducting perturbations(•) is significantly worse, but it is roughly proportional to the high-contrast case throughoutthe domain7. This means that optimal resolution meshes designed for perfectly conductingor resistive perturbations remain proper choices for reconstructing perturbations with lowercontrast since the sensitivity of each perturbation to boundary measurements remains roughlyconstant. We will make use of this observation by performing all reconstructions on meshesdesigned for perfectly conducting perturbations in section 6.

Figure 4 shows AQ as a function of the sensitivities λσ (CEM) and λ∗σ (continuum model)

at two locations: the center of the unit disk (Q=0) and a point near the boundary (Q=0.95),centered about an electrode in the CEM. In both cases, AQ depends almost linearly on thesensitivity. This means that AQ for unknown sensitivities in the CEM can be approximatedvery well by linear interpolation/extrapolation of known values.

0 0.25 0.50 0.75 10

0.010

0.020

0.030

(a) Q 7→ AQ plot for λ∗

σ = 0.015 (solid line) andfor λσ =0.015 in the CEM with 8 (H), 16 () and24 (•) electrodes.

0 0.25 0.50 0.75 10

0.010

0.020

0.030

(b) Q 7→AQ plot for λ∗

σ =0.015 (solid line) and forλσ =0.015 and for 16 electrodes with zl ≡ 10−1 (H),zl ≡ 10−2 () and zl ≡ 10−3 (•).

0 0.25 0.50 0.75 10

0.010

0.020

0.030

(c) Q 7→ AQ plot for λ∗

σ =0.015 (solid line) andfor λσ =0.015 and 16 electrodes covering 25% (H),50% () and 75% (•) of the boundary.

0 0.25 0.50 0.75 10

0.020

0.040

0.060

0.080

(d) Q 7→AQ plot for λσ =0.015 in the CEM for iso-lating (σ0 =10−6, H), conducting (σ0 =2, ) andperfectly conducting (σ0 =106, •) perturbations.The asterisks mark segment sizes of [20, Fig. 3.1]satisfying (22) for ǫ=0.1 and max|η|=1.

Figure 3. Resolved details of the continuum boundary model and the CEM forvarious settings. CEM parameters as in Figure 2 if not stated otherwise.

7In our computations shown in Figure 3d, the ratios between the areas resolved for σ0 =106 and σ0 =2 rangefrom 0.331 to 0.345 throughout the domain.


0

0.025

0.050

0.005 0.010 0.015 0.020 0.0250

0.002

0.004

0.006

Figure 4. λ∗σ 7→AQ plot (solid lines) and λσ 7→AQ plot for 8 (H,H) and 16 (,)

electrodes for a conducting (σ0 =106) perturbation at the center of a disk (Q=0,black, left ordinate) and near the boundary (Q=0.95, gray, right ordinate).

5.3. Comparison with results of MacMillan et al. [20] investigate the detectability ofperturbations in conductivity from a finite set of Neumann data for the continuum boundarymodel. The central result of [20] (cf. Corollary 2.5 and section 2.3.2. therein) is an estimate ofthe form

‖Rσiν − f‖H1/2(∂Ω) + ǫ ≥ supi1,i2 6=0

C∣∣∫

Ω(σ − σ)∇u1 · ∇u2 dx∣∣

‖i1‖H−1/2(∂Ω) ‖i2‖H−1/2(∂Ω)

, iν ∈ I, C = maxz∈Ω

σ(z)

σ(z),(21)

where I⊂H−1/2⋄ (∂Ω) is a given finite set of Neumann data, f=Rσiν+ǫn is noisy Dirichlet data,

u1 and u2 are solutions of (1),(3) for conductivity σ and Neumann data i1,i2 ∈H−1/2⋄ (∂Ω) and

ǫ= ǫ(ǫn,iν) is an error term depending on the measurement error ǫn and the non-optimality ofiν for distinguishing σ from σ. Given a fixed ǫ > 0, they generate meshes in [20, section 2.3.2.](called graded grids therein) by finding the sizes Aη = |supp(η)| of local perturbations η=σ− σsuch that

supi1,i2 6=0

C

∣∣∣∣∣

∫

supp(η)η∇u1 · ∇u2 dx

∣∣∣∣∣‖i1‖−1H−1/2(∂Ω)

‖i2‖−1H−1/2(∂Ω)

= ǫ.(22)

To compare the radial resolution of the graded grids with our results, we plot Aη versus thelocation of the center of supp(η) in Figure 3d (marked with asterisks). Lacking an explicitformula, the information was obtained from the left grid of [20, Fig. 3.1], where ǫ= 0.1 andmax|η|= 1 are considered. The perturbation sizes for a fixed ǫ, although not identical to oursolutions of section 5.2, show similar characteristics towards the boundary.

5.4. Comparison with results of a resistor network approach. The conductivity dis-cretizations in [4] arise from a model reduction approach of EIT to the problem of determiningthe resistors in a linear network [8]. The geometry of the discretization is derived from thegeometry of the resistor network which has a spiderweb structure with radial beams and con-centric rings. This geometry, while theoretically justified for near-constant radially symmetricconductivities, has the disadvantage that the number of cells in each concentric ring is constant.Thus, even though the ring segments get longer towards the center, the resolution in angulardirection gets very high at the center as the number of electrodes increases. This is not inaccordance with the loss of resolution away from the boundary. Hence the model gets unstableand sensitive to noise when increasing the number of electrodes. Figure 5 shows conductivitydiscretizations resulting from resistor networks with 13 and 25 electrodes.


Figure 5. Voronoi tessellations resulting from resistor networks with 13 (left)and 25 (right) boundary nodes. The resistor locations (marked with dots) wereobtained from the data in [4, Fig. 8 and Fig. 3]. The radial resolution near thecenter is coarse, however the angular resolution is fine due to the fixed numberof resistors in each layer.

5.5. Approximations for domains with arbitrary boundary. Most EIT applications in-volve non-circular object geometries. Even in medical applications, where e.g. the cross-sectionof a human torso is close to an ellipse, using a circular geometry for inversion introduces heavyartifacts which make it impossible to reconstruct anything meaningful if the geometry is notadjusted properly (cf. [10]). Unfortunately, analytic expressions of conformal maps and theirnormal derivatives from arbitrary simply connected domains to the unit disk are usually notavailable. Moreover, the computation of optimal resolution meshes is impractical8 as it involvesthe solution of many linear systems of equations with large dense coefficient matrices. Thus,we aim to derive a scheme for quickly generating approximations to optimal resolution mesheson arbitrary domains, motivated by the analytic results of section 5.2. There we observed thatthe sensitivities for detecting perturbations in the continuum model and the CEM are similar,but the resolution of details in the CEM is limited near the boundary (cf. Figure 3).

The idea is now to use the continuum solution rQ from (20) as an approximation for theCEM on arbitrary domains, replacing the radial coordinate Q of a point z therein by

Qz = 1− d(z), z ∈ Ω, where d(z) =

minl=1,...,L

dist(z,El)

maxζ∈Ω

minl=1,...,L

dist(ζ,El)

is the relative distance of z to the closest electrode.Inspecting Figures 3a–c, we note that the resolution of the continuum model and the CEM

have a similar characteristic, but the CEM curve is offset by the model differences and by thelimited resolution near the boundary. To account for these differences in our approximation,we add a linear correction to the continuum solution rQz which “shifts” the resolution curve tomatch the CEM results at the center and near the boundary. The linearly corrected resolutionρ should satisfy ρ(z1) = p1 and ρ(z2) = p2, where z1 and z2 are two points at which the radiip1 and p2 of resolved details of a given setting are known or can be estimated. The correctedformula reads

ρ(z) := rQz +∆p2 −∆p1Qz2 −Qz1

(Qz −Qz1) + ∆p1,(23)

where ∆p1=p1−rQz1and ∆p2=p2−rQz2

are the differences between the given setting and thecontinuum model at z1 and z2, respectively.

By (23), adaptive approximations of the optimal resolution meshes can be created withAlgorithm 1, replacing B1(O) by Ω therein and setting r= ρ(z) for each test point z ∈ T . Wewill refer to these adaptive approximations as adaptive meshes. Generating an adaptive mesh

8The computation of the meshes shown in Figure 2 involved several thousand LU decompositions with up to30 000 unknowns each and took several days (ε=0.02) to several weeks (ε=0.01) in our MATLAB implementationon a 2.2 GHz workstation with 16 CPU cores and 128 GB RAM.


that way usually takes less than one second in MATLAB on an Intel i7 notebook. For ourimplementations, we use the data from the circular case at z1 =O and z2 =(0.95,0) markedwith and in Figure 4, i.e. Qz1 =0 and Qz2 =0.95, and interpolate/extrapolate accordinglyfor a given sensitivity ε to obtain the values p1 and p2.

Figure 6 shows resulting adaptive meshes for ε=0.02 in different geometries. Figure 7 showsthe similarity of the optimal resolution mesh and its according adaptive mesh on the unit diskby plotting the centroids of the resulting Voronoi cells. Note that, as expected, the distributionand number of cells is similar (182 vs. 205 cells for ε=0.02, 445 vs. 440 for ε=0.01, cf. Figure 9).

Figure 6. Adaptive approximations of optimal resolutionmeshes in different geometries with 16 electrodes for ε=0.02.

Figure 7. Locations of theVoronoi cell centroids for theopt. res. mesh (×, 182 cells),and the adaptive mesh (•,205 cells) for ε=0.02.

5.6. Dynamic mesh size for iterative inversion algorithms. Iterative Newton-type al-gorithms successively solve forward problems and apply Newton updates to the searched-forconductivity. This means that after the kth iteration, the relative spectral error

ε(k) =

∥∥R(k) −Rmeas

∥∥2

‖Rmeas‖2can be calculated, where Rmeas is the measured ND map and R(k) is the computed ND map ofthe kth iteration for the CEM. For the k+1st iteration, we can thus generate an adaptive meshof approximate sensitivity

λσ = minα · ε(k), β

, α ∈ (0, 1), β > 0,(24)

to account for the successive refinement of reconstructed details. Then we interpolate theconductivity of the current iteration on the new mesh for the next iteration. This dynamicrefinement is particularly helpful for generating meshes when the spectral error level of themeasured data is unknown.

In our tests, we choose α=0.8, β=0.05 and only generate a new mesh in the klth iterationif the relative spectral error decreased sufficiently, i.e. if ε(kl)≤α ·ε(kl−1), where kl−1 is the lastiteration where an adaptive mesh was generated. The dynamic refinement during a reconstruc-tion with εmeas=0.25% shown in Figure 8. If the number of conductivity coefficients should belimited for computational reasons, a lower limit for λσ could be added to (24).

6. Numerical results

In this section we present the performance of optimal resolution meshes and its adaptiveapproximations for the reconstruction of EIT images. The algorithm used is the regularizedinexact Newton method REGINN described in [25] and applied to EIT in [18]. In the conjugategradient iteration of REGINN, we use the Euclidean inner product on R

n for the discreteconductivity space instead of the area-dependent L2 inner product used in [18]. This is because


(a) k=1, 76 cells (b) k=8, 274 cells (c) k=13, 885 cells (d) k=25, 1992 cells

Figure 8. Adaptive meshes generated in kth iteration of an inversion.

by design, the impact of each conductivity segment to measured data is roughly the same,independent of its area. Note that REGINN performs absolute image reconstructions, i.e. noreference data of the same setting with a homogeneous background are needed.

For the simulated data in section 6.1 and 6.2, M =L measurements are given by the so-calledadjacent current patterns

I(m)k =

1, k = m,

−1, k = 1 + (mmodL) ,

0, otherwise.

for m,k = 1, ... ,L. When adding artificial noise of level δ > 0 to the corresponding potentialvectors U (j) ∈R

L, we mean component-wise relative noise

U(j)noisy = U (j) + γj

(n(j)1 U

(j)1 , . . . , n

(j)L U

(j)L

)⊤,

where n(j) ∈ [−1,1]L is uniformly distributed noise and γj > 0 is chosen such that∥∥∥U (j)

noisy − U (j)∥∥∥2= δ, j = 1, . . . ,M.

Note that in general δ 6= ε for the relative spectral error ε of the resulting noisy ND map. Therelative error between a reconstructed conductivity σrec and the exact solution σ is computed as

e =‖P∆σrec − P∆σ‖L2(∆)

‖P∆σ‖L2(∆)

,

where P∆ is an interpolation on a very fine triangulation ∆ of the domain Ω. Throughoutall numerical simulations, we use L= 16 equally distributed electrodes that cover 50% of theboundary with contact impedances zl =0.01 2π

|∂Ω| , l=1,...,L.

All forward computations are performed using finite elements on 10 000–13 000 triangles whichare refined near the electrodes and independent of the conductivity discretizations. For example,the forward mesh for all reconstructions of Figure 9 consists of 12 037 triangles, see Figure 11.

The REGINN parameters are µ0 = 0.6, ζ = 0.95 and τ = 1.05 (denoted R in [18, section 6])which we found to be robust throughout all our experiments. Thus, the noise level is the onlyregularization parameter that needs to be specified when reconstructing from measured data.

REGINN terminates according to a discrepancy principle, that is after nit iterations, wherenit is the first integer satisfying

maxj=1,...,M

∥∥∥U (j)meas −R(nit)I

(j)∥∥∥2≤ τδ < max

j=1,...,M

∥∥∥U (j)meas −R(n)I

(j)∥∥∥2, n = 1, . . . , nit,(25)

for a given measurement set of Current-Voltage pairs (I(j),U(j)meas), j=1,...,M , M ∈N.

We observed that the number of iterations until convergence significantly depends on the errorlevel δ and the type of inclusions. For conducting inclusions and δ=1%, we have nit ≈ 5–15.For resistive inclusions and δ = 0.25%, nit ≈ 50–80. Each iteration takes less than one secondin MATLAB on an Intel i7 notebook.


6.1. Reconstructions on the unit disk. First, we reconstruct circular inclusions of conduc-tivity σ0=0.5 and radius rQ=0.4 inside the unit disk with background conductivity σ1=0.25.As we assume the background conductivity to be unknown, the initial guess is σinit≡ 0.4 on Ω.To avoid inverse crime, the data Rσ is generated with the analytic forward solver of section 4.

To demonstrate the performance of the sensitivity-based meshes, reconstructions are alsoperformed on generic meshes with approximately the same number of cells for comparison.Figure 9 shows reconstructions on an optimal resolution mesh (b), an adaptive approximationmesh (c), a generic triangle mesh (see Figure 10) which is refined near the boundary generatedby the MATLAB PDE toolbox (d) and a uniformly spaced mesh (e). All inverse problemsare underdetermined as the number of conductivity coefficients exceeds the upper bound ofL(L−1)

2 = 120 degrees of freedom of an L= 16 electrode setting. All inversions are terminatedby criterion (25), but using adaptive meshes results in faster convergence, smaller conductivityerrors and fewer visible artifacts.

(a) Trueconductivity

nit=8, e=21.8%

nit=10, e=16.2%

(b) Opt.res. mesh,(445 cells)

nit=9, e=22.2%

nit =12, e=16.5%

(c) Adapt. mesh,(440 cells)

nit =10, e=22.7%

nit =16, e=17.4%

(d) Triangle mesh(452 cells)

nit=21, e=24.7%

nit=42, e=18.8%

(e) Uniform mesh(447 cells)

0

0.1

0.2

0.3

0.4

0.5

Figure 9. Settings with conducting inclusions and reconstructions from noisydata (δ=1%). The refinement parameters of the triangle mesh and the uniformmesh were adjusted “by hand” to get a cell count as close as possible to the opt.res. mesh.

Figure 10. Generic 452 triangle meshgenerated by the MATLAB PDE tool-box for the reconstruction of Figure 9d.

Figure 11. FEM discretization with12 037 triangles for solving the forwardproblem on the unit disk, see Figure 9.


The conductivity discretization affects the representation of the Frechet derivative of themap σ 7→ Rσ used in the Newton algorithm. This impact is shown in Figure 12, where theFrobenius norms of the Jacobians in each canonical direction χΩk

, k=1,...,K, are plotted indescending order for the partitions Ω=

⋃k=1,...,KΩk of the corresponding discretizations. It is

not surprising that the sensitivity-based discretizations lead to a better equilibration of energyacross the Frechet derivative in the canonical basis. This type of conditioning positively affectsthe reconstruction quality and the speed of convergence of the Newton algorithm. In contrast,the conductivity artifacts near the electrodes in the uniform discretization appear on segmentswith particularly high sensitivity. A thorough analysis of this observation is beyond the scopeof this work and subject to future investigation.

0 50 100 150 200 250 300 350 400 4500

0.2

0.4

0.6

0.8

1

1.2

Figure 12. Distribution of energy of the Frechet derivative. The plot shows the Frobeniusnorms of the Frechet derivative in the canonical directions in descending order for the opti-mal resolution mesh (red solid), the adaptive mesh (blue dashed), the triangle mesh (greendotted-dashed) and the uniform mesh (black dotted). The optimal resolution mesh resultsin a more equilibrated Jacobian for the Newton algorithm, leading to faster convergence.

6.2. Reconstructions on polygonal domains. Next, we demonstrate the performance ofadaptive meshes on non-circular domains (optimal resolution meshes cannot be computed withthe given analytic solution for non-circular domains). The simulated data is generated by finiteelements on very fine triangle meshes (≈30000) which are not refinements of the forward meshesto avoid inverse crime. The initial guess for all settings is σinit≡ 0.3. The results are shown inFigure 13.

The first setting is a square with conductivity σ=0.25, a conducting segment (σ=0.5), aresistive segment (σ=0.1) and δ=1% noise. The challenge of this setting are the jumps ofconductivity in the interior and along the boundary.

The second setting is a chest-shaped domain with σ=0.25, highly resistive inclusions (σ=0.05),a conducting inclusion (σ=0.5) and δ=0.5% noise. The small conductivity coefficient in largeparts of the domain means a small ellipticity constant of the variational formulation, leading toslower convergence.


The third setting is an L-shaped domain with σ=0.25, a conducting circular inclusion(σ=0.5), a resistive annulus (σ=0.1) and δ=0.25% noise. Note that the small noise level leadsto very fine adaptive conductivity meshes, in particular for elongated non-convex domains.

δ=1%

δ=0.5%

δ=0.25%

(a) Setting

887 cells, nit=63,e=17.8%

2152 cells, nit=60,e=19.6%

5126 cells, nit=26,e=15.5%

(b) Adaptive mesh

888 cells,nit=181,e=20.8%

2152 cells, nit=198,e=20.1%

5126 cells, nit=102,e=18.5%

(c) Triangle mesh

900 cells,nit=414,e=21.1%

2152 cells, nit=124,e=21.3%

5126 cells, nit=84,e=17.1%

(d) Uniform mesh

0

0.1

0.2

0.3

0.4

0.5

Figure 13. Reconstructions from noisy simulated data on adaptive meshes,generic triangle meshes and uniform meshes.

6.3. Reconstructions from measured data. The measured data from a tank experiment waskindly provided by Aku Seppanen (University of Eastern Finland) and Stratos Staboulis (AaltoUniversity). The tank has a circumference of 88 cm and a height of 7 cm, each of the 16 electrodeshas a width of 2.5 cm. It is filled with saline (0.016Ω−1m−1) and two conducting (metal) objects.The data was measured with the Kuopio Impedance Tomography 4 device, see [17]. Low-frequency alternating currents (1 kHz, 1mA) were injected between a fixed grounded “driving”electrode (north-east in Figure 14) and each of the other 15 electrodes in turn (thus M =L−1),ignoring the phase information of the applied currents and the resulting potentials. By averagingover multiple independent measurements, an estimated measurement tolerance of δ≈ 0.1–0.2%was achieved. To account for the imperfections of the model (2D reconstruction of a cylindrical3D setting, exact electrode positions) and contact impedances (we assume zl =0.01 2π

|∂Ω|), we set

δ=0.3% as the estimated measurement error for the REGINN algorithm. The reconstructionsare shown in Figure 14. Due to the small error level, the computation of an according optimalresolution mesh is not feasible, cf. section 5.5.

7. Conclusions

Using conformal maps, we introduced an analytic method to determine the sensitivity ofboundary measurements to circular perturbations in conductivity for the CEM on a disk. With


Saline tank withmetal inclusions

Adaptive mesh

1193 cells, nit=26

Triangle mesh

1194 cells, nit=30

Uniform mesh

1193 cells, nit=46 0

0.1

0.2

0.3

0.4

0.5

Figure 14. Measurement setting and reconstructions from measured tank data.

this information, we quantified the spatial resolution of various CEM settings and discretizedthe conductivity space accordingly, resulting in improved robustness and faster convergencewhen solving the ICP with Newton-type algorithms and noisy data.

Moreover, we pointed out the connections and differences of the CEM and the continuumboundary model in terms of imaging resolution.

Finally, we derived a heuristic approximation of sensitivity based discretizations of the con-ductivity space for circular and non-circular domains and verified its performance with simulatedand measured data.

8. Acknowledgements

The authors thank Nuutti Hyvonen9 and Stratos Staboulis9 for valuable insights and fruitfuldiscussions on the Complete Electrode Model. We are grateful for the tank data provided byAku Seppanen10 and Stratos Staboulis.

Moreover, we are indebted to the anonymous referees. Their thoughtful suggestions greatlyhelped us to improve the presentation of our results.

Appendix A. Computing the analytic solution

Here we derive the formulas for the analytic forward solution of the CEM.

A.1. Fourier representation of the potential for a centered perturbation. We willexpress the potential u= u(r,θ) on Ω as a Fourier series in polar coordinates. The potential ucan be split in even and odd parts in θ and each part can be treated separately, hence assumefirst that u is even in θ. According to [29, (A3.4)], u is of the form

u(r, θ) = u0 +

∞∑

k=1

(r

r0

)k

a(1)k cos(kθ), r ≤ r0,

∞∑

k=1

(r−ka

(2)k + rka

(3)k

)cos(kθ), r0 ≤ r < 1,

for some coefficients a(1,2,3)k ∈ R. By matching u and σ∂u/∂r at r = r0, we find from the

orthogonality of the Fourier basis that

a(1)k = a

(3)k rk0 + a

(2)k r−k

0 (condition on u) and

σ0a(1)k = σ1

(a(3)k rk0 − a

(2)k r−k

0

)(condition on σ∂u/∂r) .

Substituting the first into the second equation, we get

a(2)k = cka

(3)k with ck :=

σ1/σ0 − 1

σ1/σ0 + 1r2k0 ,

9Aalto University, Helsinki, Finland.10University of Eastern Finland, Kuopio, Finland.


thus outside the perturbation, u has the representation

u(r, θ) = u0 +

∞∑

k=1

(rk + r−kck

)a(3)k cos(kθ),

∂u

∂r(r, θ) =

∞∑

k=1

k(rk−1 − r−k−1ck

)a(3)k cos(kθ).

By evaluating u at r=1 and letting ak := (1+ck)a(3)k , we can express the potential and normal

current at the boundary as

u(1, θ) = u0 +

∞∑

k=1

ak cos(kθ),

σ1∂u

∂r(1, θ) = σ1

∞∑

k=1

dk ak cos(kθ), where dk := k1− ck1 + ck

.

For the odd part of u in θ we can do the same calculations, replacing the cosine terms by sineterms and introducing coefficients bk. Combining both, we get the desired representations (10).

A.2. Computation of the Fourier coefficients for given electrode potentials. We nowapply the approach of [11, Appendix] to (10) to determine the Fourier coefficients u0 and ak,bk,k∈N, therein for a given voltage pattern U =(U1,...,UL)

⊤ in the CEM. To that end, we rewrite(4b) as iν =

1zl(Ul−u) and plug in the representations (10) for u and iν which yields

σ1

∞∑

k=1

dk (ak cos(kθ) + bk sin(kθ))

=

1zl(Ul − u0 −

∑∞k=1 ak cos(kθ) + bk sin(kθ)) on El, l = 1, . . . , L,

0, otherwise.(26)

Following the idea of [11,Appendix], we multiply (26) with

cos(nθ), n ∈ N0, and sin(nθ), n ∈ N,

respectively and integrate in θ over [0,2π] which leads to the set of equations

0 =

L∑

l=1

Ul − u0zl

2ωl

−∞∑

k=1

akzl

∫ θl+ωl

θl−ωl

cos(kθ) dθ +bkzl

∫ θl+ωl

θl−ωl

sin(kθ) dθ for n = 0,

σ1πdnan =L∑

l=1

Ul − u0zl

∫ θl+ωl

θl−ωl

cos(nθ) dθ

−∞∑

k=1

akzl

∫ θl+ωl

θl−ωl

cos(nθ) cos(kθ) dθ +bkzl

∫ θl+ωl

θl−ωl

cos(nθ) sin(kθ) dθ, n ∈ N,

σ1πdnbn =

L∑

l=1

Ul − u0zl

∫ θl+ωl

θl−ωl

sin(nθ) dθ

−∞∑

k=1

akzl

∫ θl+ωl

θl−ωl

sin(nθ) cos(kθ) dθ +bkzl

∫ θl+ωl

θl−ωl

sin(nθ) sin(kθ) dθ, n ∈ N,

(27)


where θl ∈ [0,2π) is the angular coordinate of the lth electrode center and ωl > 0 is its angularhalf-width. These equations can be rewritten as an infinite system of linear equations foru0,ak,bk as

(rU

sU

)=

(A B

B⊤ C

)(u0, a1, a2, . . . , b1, b2, . . .

)⊤,(28)

with the (infinite dimensional) vectors

rU =(rU0 , r

U1 , . . .

)⊤, sU =

(sU1 , s

U2 , . . .

)⊤

and matrices

A =

A00 A01 . . .A10 A11 . . .

......

. . .

, B =

B01 B02 . . .B11 B12 . . .

......

. . .

, C =

C11 C12 . . .C21 C22 . . .

......

. . .

with entries

Ank =L∑

l=1

1

zl

∫ θl+ωl

θl−ωl

cos(nθ) cos(kθ) dθ

︸︷︷︸= 1

2[sl(k−n)+sl(k+n)]

+δnkσ1πdk, n ∈ N0, k ∈ N0,(29)

Bnk =L∑

l=1

1

zl

∫ θl+ωl

θl−ωl

cos(nθ) sin(kθ) dθ

︸︷︷︸=− 1

2[cl(k−n)+cl(k+n)]

, n ∈ N0, k ∈ N,

Cnk =L∑

l=1

1

zl

∫ θl+ωl

θl−ωl

sin(nθ) sin(kθ) dθ

︸︷︷︸= 1

2[sl(k−n)−sl(k+n)]

+δnkσ1πdk, n ∈ N, k ∈ N,

rUn =L∑

l=1

Ul

zl

∫ θl+ωl

θl−ωl

cos(nθ) dθ

︸︷︷︸=sl(n)

, n ∈ N0,

sUn =L∑

l=1

Ul

zl

∫ θl+ωl

θl−ωl

sin(nθ) dθ

︸︷︷︸=−cl(n)

, n ∈ N,

where δnk is the Kronecker delta. Above integrals have analytic solutions given by the expres-sions sl and cl, where sl(0)= 2ωl, cl(0)= 0 and

sl(n) =sin(n(θl+ωl))−sin(n(θl−ωl))

n , cl(n) =cos(n(θl+ωl))−cos(n(θl−ωl))

n , n ∈ Z \ 0 .Once the Fourier coefficients are known, we can compute the lth entry of the resulting currentvector I by (4c), integrating the left-hand side of (26) over the lth electrode:

Il =

∫

El

iν dS = σ1

∞∑

k=1

dk (aksl(k) − bkcl(k)) .

However, it is advisable to integrate over the faster converging Fourier series on the right-handside of (26) which yields

Il =1

zl

(2ωl(Ul − u0)−

∞∑

k=1

aksl(k)− bkcl(k)

).


Remark : In practice, we truncate the Fourier series to get a finite system of linear equations.The truncation index N should be chosen with respect to the electrode widths such that thepotential and normal current along all electrodes are approximated well by the truncated Fourierseries. For example, if 2ωmin is the smallest electrode angular width, the truncation index shouldbe chosen well above the “critical” index ⌈2π/(2ωmin)⌉ of a Fourier sum that can resolve detailsof size 2ωmin. In our implementations, we found N =max⌈32π/ωmin⌉,1000 to give results ofvery high accuracy.

Appendix B. Analytic solution for the CEM forward problem under conformal

mapping

By the Riemann mapping theorem, any simply connected smooth domain can be mapped tothe unit circle conformally, that is, angle-preserving [23, 26]. We use conformal maps to reducenon-concentric EIT geometries to the concentric circular case. In this simpler geometry, wecan solve the forward problem in EIT analytically. We now derive the analytic solution for thetransformed CEM forward problem, incorporating the non-constant contact impedances in theformulas of Appendix A.2. Since the reciprocal of the transformed conductivity given in (18)is unsuitable for analytic (closed form) integration when multiplied by cosine or sine terms, werewrite equation (19b) to avoid a term of the form 1

cos(ϑ) on either side of the equation. To

achieve this, we use the representation

1 +R2

1−R2iwν (ϑ) =

1

zl(Ul − fw(ϑ))− 2R cos(ϑ)

1−R2iwν (ϑ) on Ew

l ,

0 otherwise.(30)

As in the centered case, we multiply the Fourier representation of (30) with the test functionscos(nϑ), n ∈ N0, and sin(nϑ), n ∈ N, respectively and integrate in ϑ over [0,2π]. For betterreadability, we write t(nϑ) for any of these test functions. On the left-hand side, we get

∫ 2π

0t (nϑ)

1 +R2

1−R2iwν (ϑ) dϑ =

0 for n = 0,

πσ11 +R2

1−R2dnan for the cosine terms,

πσ11 +R2

1−R2dnbn for the sine terms.

(31)

On the right-hand side, we get

L∑

l=1

Ul

zl

∫ ϑl+vl

ϑl−vl

t (nϑ) dϑ− u0

L∑

l=1

1

zl

∫ ϑl+vl

ϑl−vl

t (nϑ) dϑ

−∞∑

k=1

ak

L∑

l=1

[1

zl

∫ ϑl+vl

ϑl−vl

t (nϑ) cos(kϑ) dϑ +2Rσ1dk1−R2

∫ ϑl+vl

ϑl−vl

t (nϑ) cos(ϑ) cos(kϑ) dϑ

]

−∞∑

k=1

bk

L∑

l=1

[1

zl

∫ ϑl+vl

ϑl−vl

t (nϑ) sin(kϑ) dϑ +2Rσ1dk1−R2

∫ ϑl+vl

ϑl−vl

t (nϑ) cos(ϑ) sin(kϑ) dϑ

].

Now we rearrange each equation to be used as one row of a linear system of equations for u0and ak,bk, k∈N, and return to the notation of (28). In the conformally mapped case, we get

(rU

sU

)=

(A B1

B2 C

)(u0, a1, a2, . . . b1, b2, . . .

)⊤.(32)


The matrix coefficients are

Ank =L∑

l=1

1

zl

∫ ϑl+vl

ϑl−vl

cos(nϑ) cos(kϑ) dϑ+ δnkπσ11 +R2

1−R2dk

+2Rσ1dk1−R2

L∑

l=1

∫ ϑl+vl

ϑl−vl

cos(nϑ) cos(ϑ) cos(kϑ) dϑ, n ∈ N0, k ∈ N0,

B1nk =

L∑

l=1

1

zl

∫ ϑl+vl

ϑl−vl

cos(nϑ) sin(kϑ) dϑ

+2Rσ1dk1−R2

L∑

l=1

∫ ϑl+vl

ϑl−vl

cos(nϑ) cos(ϑ) sin(kϑ) dϑ, n ∈ N0, k ∈ N,

B2nk =

L∑

l=1

1

zl

∫ ϑl+vl

ϑl−vl

sin(nϑ) cos(kϑ) dϑ

+2Rσ1dk1−R2

L∑

l=1

∫ ϑl+vl

ϑl−vl

sin(nϑ) cos(ϑ) cos(kϑ) dϑ, n ∈ N, k ∈ N0,

Cnk =

L∑

l=1

1

zl

∫ ϑl+vl

ϑl−vl

sin(nϑ) sin(kϑ) dϑ+ δnkπσ11 +R2

1−R2dk

+2Rσ1dk1−R2

L∑

l=1

∫ ϑl+vl

ϑl−vl

sin(nϑ) cos(ϑ) sin(kϑ) dϑ, n ∈ N, k ∈ N,

rn =L∑

l=1

Ul

zl

∫ ϑl+vl

ϑl−vl

cos(nϑ) dϑ, n ∈ N0,

sn =

L∑

l=1

Ul

zl

∫ ϑl+vl

ϑl−vl

sin(nϑ) dϑ. n ∈ N.

Note in particular that B1nk 6=B2

kn in general for R> 0 due to the influence of the coefficientsdk. In this representation, all integrals are given analytically by trigonometric identities. Theintegrals without the cos(ϑ) factor match those of (29). The analytic solutions of the otherintegrals read

1

4

[sl(αn,k) + sl(βn,k) + sl(γn,k) + sl(κn,k)

]in Ank,

1

4

[cl(αn,k) + cl(βn,k)− cl(γn,k)− cl(κn,k)

]in B1

nk and B2kn,

1

4

[−sl(αn,k) + sl(βn,k) + sl(γn,k)− sl(κn,k)

]in Cnk,

where

αn,k = −k − n+ 1, βn,k = −k + n+ 1, γn,k = k − n+ 1, κn,k = k + n+ 1.

The solution of this system is the set of Fourier coefficients of the transformed Dirichlet datafw. For the truncation and the computation of the current vector, the same comments applyas in Appendix A.2. In particular, it is advisable to integrate over the right-hand side of (30)


which yields

Il =1−R2

(1 +R2)zl

(2ωl(Ul − u0)−

∞∑

k=1

aksl(k)− bkcl(k)

)

+Rσ1

1 +R2

∞∑

k=1

dk [ak (sl(k − 1) + sl(k + 1))− bk (cl(k − 1) + cl(k + 1))] .

Appendix C. Monotony of the sensitivity

This is a CEM version of [13, Appendix I]. The energy functional

J(u,U) =1

2a ((u,U), (u,U)) −

L∑

l=1

IlUl

for the bilinear operator a from (6) has the minimizing property

J(u∗, U∗) = min(u,U)∈H1(Ω)×RL

J(u,U),

where (u∗,U∗) is the solution of the CEM for current vector I. Denote by aB and aD thebilinear operators, by JB and JD the energy functionals and by (uB ,UB) and (uD,UD) theCEM solutions for conductivities σB and σD from (12), respectively. Using (5), we immediatelyget

JB(uB , UB) = −1

2I⊤UB and JD(uD, UD) = −1

2I⊤UD.

From σD ≥σB on Ω, it follows that

JD(uD, UD)− JB(uD, UD) =1

2

∫

Ω(σD − σB) |∇uD|2 dx ≥ 0.

Using the minimizing property for JB , we get

−I⊤UD = 2JD(uD, UD) ≥ 2JB(uD, UD) ≥ 2JB(uB , UB) = −I⊤UB.

In terms of ND maps, we have that UB =RσBI and UD =RσD

I, thus

〈I, (RσD−RσB

)I〉 ≤ 0

for all 0 6= I ∈RL⋄ . Similarly, we get

〈I, (RσD−Rσ1

)I〉 ≤ 0 and 〈I, (RσB−Rσ1

)I〉 ≤ 0

and finally 〈I,(RσD−Rσ1

)I〉≤ 〈I,(RσB−Rσ1

)I〉≤ 0, hence λσD,σ1≥λσB ,σ1

.

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Institute for Applied and Numerical Mathematics,

Karlsruhe Institute of Technology, DE-76049 Karlsruhe, Germany.

Documents

RESOLUTION-CONTROLLED CONDUCTIVITY ...winkler/media/res_eit.pdfCEM in [14]. For σ,eσ∈L ∞ + (Ω) and considering Rσ as an operator on L2(∂Ω), we call λσ,eσ = R σ,L −Reσ,L